{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are several common variants of the array variance calculation. Assuming the input " }, { "__type": "CrossRef", "__tag": 4002, "value": "a", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "a" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is a one-dimensional NumPy array and " }, { "__type": "InlineCode", "__tag": 4051, "value": "mean" }, { "__type": "Text", "__tag": 4046, "value": " is either provided as an argument or computed as " }, { "__type": "InlineCode", "__tag": 4051, "value": "a.mean()" }, { "__type": "Text", "__tag": 4046, "value": ", NumPy computes the variance of an array as " } ] }, { "__type": "Code", "__tag": 4050, "value": "N = len(a)\nd2 = abs(a - mean)**2 # abs is for complex `a`\nvar = d2.sum() / (N - ddof) # note use of `ddof`", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Different values of the argument " }, { "__type": "CrossRef", "__tag": 4002, "value": "ddof", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "ddof" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " are useful in different contexts. NumPy's default " }, { "__type": "InlineCode", "__tag": 4051, "value": "ddof=0" }, { "__type": "Text", "__tag": 4046, "value": " corresponds with the expression:" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\frac{\\sum_i{|a_i - \\bar{a}|^2 }}{N}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "which is sometimes called the \"population variance\" in the field of statistics because it applies the definition of variance to " }, { "__type": "CrossRef", "__tag": 4002, "value": "a", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "a" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " as if " }, { "__type": "CrossRef", "__tag": 4002, "value": "a", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "a" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " were a complete population of possible observations." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Many other libraries define the variance of an array differently, e.g.:" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\frac{\\sum_i{|a_i - \\bar{a}|^2}}{N - 1}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In statistics, the resulting quantity is sometimes called the \"sample variance\" because if " }, { "__type": "CrossRef", "__tag": 4002, "value": "a", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "a" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is a random sample from a larger population, this calculation provides an unbiased estimate of the variance of the population. The use of " }, { "__type": "InlineMath", "__tag": 4057, "value": "N-1" }, { "__type": "Text", "__tag": 4046, "value": " in the denominator is often called \"Bessel's correction\" because it corrects for bias (toward lower values) in the variance estimate introduced when the sample mean of " }, { "__type": "CrossRef", "__tag": 4002, "value": "a", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "a" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is used in place of the true mean of the population. For this quantity, use " }, { "__type": "InlineCode", "__tag": 4051, "value": "ddof=1" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that for complex numbers, the absolute value is taken before squaring, so that the result is always real and nonnegative." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For floating-point input, the variance is computed using the same precision the input has. Depending on the input data, this can cause the results to be inaccurate, especially for " }, { "__type": "CrossRef", "__tag": 4002, "value": "float32", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:float32" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " (see example below). Specifying a higher-accuracy accumulator using the " }, { "__type": "InlineCode", "__tag": 4051, "value": "dtype" }, { "__type": "Text", "__tag": 4046, "value": " keyword can alleviate this issue." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "variance", "annotation": "ndarray, see dtype parameter above", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "out=None" }, { "__type": "Text", "__tag": 4046, "value": ", returns a new array containing the variance; otherwise, a reference to the output array is returned." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute the variance along the specified axis." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "a", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array containing numbers whose variance is desired. If " }, { "__type": "CrossRef", "__tag": 4002, "value": "a", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "a" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is not an array, a conversion is attempted." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "axis", "annotation": "None or int or tuple of ints, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Axis or axes along which the variance is computed. The default is to compute the variance of the flattened array. If this is a tuple of ints, a variance is performed over multiple axes, instead of a single axis or all the axes as before." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "dtype", "annotation": "data-type, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Type to use in computing the variance. For arrays of integer type the default is " }, { "__type": "CrossRef", "__tag": 4002, "value": "float64", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:float64" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "; for arrays of float types it is the same as the array type." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "out", "annotation": "ndarray, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Alternate output array in which to place the result. It must have the same shape as the expected output, but the type is cast if necessary." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "ddof", "annotation": "{int, float}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "\"Delta Degrees of Freedom\": the divisor used in the calculation is " }, { "__type": "InlineCode", "__tag": 4051, "value": "N - ddof" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineCode", "__tag": 4051, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " represents the number of elements. By default " }, { "__type": "CrossRef", "__tag": 4002, "value": "ddof", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "ddof" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is zero. See notes for details about use of " }, { "__type": "CrossRef", "__tag": 4002, "value": "ddof", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "ddof" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "keepdims", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If the default value is passed, then " }, { "__type": "CrossRef", "__tag": 4002, "value": "keepdims", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "keepdims" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " will not be passed through to the " }, { "__type": "CrossRef", "__tag": 4002, "value": "var", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:var" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " method of sub-classes of " }, { "__type": "CrossRef", "__tag": 4002, "value": "ndarray", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:ndarray" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", however any non-default value will be. If the sub-class' method does not implement " }, { "__type": "CrossRef", "__tag": 4002, "value": "keepdims", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "keepdims" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " any exceptions will be raised." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "where", "annotation": "array_like of bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Elements to include in the variance. See " }, { "__type": "InlineRole", "__tag": 4003, "value": "~numpy.ufunc.reduce", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for details." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.20.0" } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "mean", "annotation": "array like, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Provide the mean to prevent its recalculation. The mean should have a shape as if it was calculated with " }, { "__type": "InlineCode", "__tag": 4051, "value": "keepdims=True" }, { "__type": "Text", "__tag": 4046, "value": ". The axis for the calculation of the mean should be the same as used in the call to this var function." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 2.0.0" } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "correction", "annotation": "{int, float}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array API compatible name for the " }, { "__type": "InlineCode", "__tag": 4051, "value": "ddof" }, { "__type": "Text", "__tag": 4046, "value": " parameter. Only one of them can be provided at the same time." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 2.0.0" } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Returns the variance of the array elements, a measure of the spread of a distribution. The variance is computed for the flattened array by default, otherwise over the specified axis." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/numpy/ma/core.py", "item_line": 7063, "item_type": "function", "aliases": [ "numpy.ma.var" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\na = np.array([[1, 2], [3, 4]])\nnp.var(a)\nnp.var(a, axis=0)\nnp.var(a, axis=1)\n", "execution_status": "none" }, { "__type": "Text", "__tag": 4046, "value": "\nIn single precision, var() can be inaccurate:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "a = np.zeros((2, 512*512), dtype=np.float32)\na[0, :] = 1.0\na[1, :] = 0.1\nnp.var(a)\n", "execution_status": "none" }, { "__type": "Text", "__tag": 4046, "value": "\nComputing the variance in float64 is more accurate:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "np.var(a, dtype=np.float64)\n((1-0.55)**2 + (0.1-0.55)**2)/2\n", "execution_status": "none" }, { "__type": "Text", "__tag": 4046, "value": "\nSpecifying a where argument:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "a = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])\nnp.var(a)\nnp.var(a, where=[[True], [True], [False]])\n", "execution_status": "none" }, { "__type": "Text", "__tag": 4046, "value": "\nUsing the mean keyword to save computation time:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom timeit import timeit\na = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])\nmean = np.mean(a, axis=1, keepdims=True)\ng = globals()\nn = 10000\nt1 = timeit(\"var = np.var(a, axis=1, mean=mean)\", globals=g, number=n)\nt2 = timeit(\"var = np.var(a, axis=1)\", globals=g, number=n)\nprint(f'Percentage execution time saved {100*(t2-t1)/t2:.0f}%')\n", "execution_status": "none" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "mean", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:mean" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "nanmean", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:nanmean" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "nanstd", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:nanstd" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "nanvar", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:nanvar" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "std", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:std" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "ufuncs-output-type", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "ufuncs-output-type" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "ref" } ] } ], "type": null } ], "signature": null, "references": null, "qa": "numpy.ma.core:var", "arbitrary": [], "local_refs": [ "a", "axis", "correction", "ddof", "dtype", "keepdims", "mean", "out", "variance", "where" ] }